# Difference between revisions of "2006 AMC 10A Problems/Problem 10"

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<math> \mathrm{(A) \ } 3\qquad \mathrm{(B) \ } 6\qquad \mathrm{(C) \ } 9\qquad \mathrm{(D) \ } 10\qquad \mathrm{(E) \ } 11 </math> | <math> \mathrm{(A) \ } 3\qquad \mathrm{(B) \ } 6\qquad \mathrm{(C) \ } 9\qquad \mathrm{(D) \ } 10\qquad \mathrm{(E) \ } 11 </math> | ||

== Solution == | == Solution == | ||

− | Since <math>\sqrt{x}</math> cannot be negative, the | + | Since <math>\sqrt{x}</math> cannot be negative, the outermost [[radicand]] is at most 120. We are interested in the number of integer values that the expression can take, so we count the number of squares less than 120, the greatest of which is <math>10^2=100.</math> |

Thus our set of values is | Thus our set of values is | ||

− | <center><math> \{ | + | <center><math> \{10^2, 9^2,\ldots,2^2, 1^2, 0^2\} </math></center> |

And our answer is '''11, (E)''' | And our answer is '''11, (E)''' |

## Revision as of 01:36, 28 February 2007

## Problem

For how many real values of is an integer?

## Solution

Since cannot be negative, the outermost radicand is at most 120. We are interested in the number of integer values that the expression can take, so we count the number of squares less than 120, the greatest of which is

Thus our set of values is

And our answer is **11, (E)**